The question of which department should start the invoice process really depends on the company. It could be sales, which directly contacts the client, it could be production, which knows when the final delivery is made, or it could be accounting, which controls all orders. The below workflow sample has the production department start the invoice issuance process once the product is delivered.
Of course, depending on the client's satisfaction and what-not after delivery, the production team may be too busy to remember to start the process. In this case, accounting can go ahead and start the process ahead of time, in other words add the task in production's to-do list.
By the way, complete transparency of invoice processes are often difficult due to IR issues, but you do want to let as many employees as possible see them.